• ETRI Journal
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• v.36 no.6
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• pp.1032-1040
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• 2014

The Lai-Massey scheme, proposed by Vaudenay, is a modified structure in the International Data Encryption Algorithm cipher. A family of block ciphers, named FOX, were built on the Lai-Massey scheme. Impossible differential cryptanalysis is a powerful technique used to recover the secret key of block ciphers. This paper studies the impossible differential cryptanalysis of the Lai-Massey scheme with affine orthomorphism for the first time. Firstly, we prove that there always exist 4-round impossible differentials of a Lai-Massey cipher having a bijective F-function. Such 4-round impossible differentials can be used to help find 4-round impossible differentials of FOX64 and FOX128. Moreover, we give some sufficient conditions to characterize the existence of 5-, 6-, and 7-round impossible differentials of Lai-Massey ciphers having a substitution-permutation (SP) F-function, and we observe that if Lai-Massey ciphers having an SP F-function use the same diffusion layer and orthomorphism as a FOX64, then there are indeed 5- and 6-round impossible differentials. These results indicate that both the diffusion layer and orthomorphism should be chosen carefully so as to make the Lai-Massey cipher secure against impossible differential cryptanalysis.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.8 no.10
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• pp.3624-3637
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• 2014

At SAC 2004, Junod and Vaudenay designed the FOX family based on the Lai-Massey scheme. They noted that it was impossible to find any useful differential characteristic or linear trail after 8 rounds of FOX64 or FOX128. In this paper, we provide the lower bound of differentially active S-boxes in consecutive rounds of the Lai-Massey scheme that has SPS as its F-function, and we propose the necessary conditions for the reachability of the lower bound. We demonstrate that similar results can be obtained with respect to the lower bound of linearly active S-boxes by proving the duality in the Lai-Massey scheme. Finally, we apply these results to FOX64 and FOX128 and prove that it is impossible to find any useful differential characteristics or linear trail after 6 rounds of FOX64. We provide a more precise security bound for FOX128.